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Kinetic and Hydrodynamic Models for Multi-Band Quantum Transport in Crystals

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Multi-Band Effective Mass Approximations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 94))

Abstract

This chapter is devoted to the derivation of k⋅p multi-band quantum transport models, in both the pure-state and mixed-state cases. The first part of the chapter deals with pure-states. Transport models are derived from the crystal periodic Hamiltonian by assuming that the lattice constant is small, so that an effective multi-band Schrödinger equation can be written for the envelopes of the wave functions of the charge carriers. Two principal approaches are presented here: one is based on the Wannier-Slater envelope functions and the other on the Luttinger-Kohn envelope functions. The concept of Wannier functions is then generalized, in order to study the dynamics of carriers in crystals with varying composition (heterostructures). Some of the most common approximations, like the single band, mini-bands and semi-classical transport, are derived as a limit of multi-band models. In the second part of the chapter, the mixed-state (i.e. statistical) case is considered. In particular, the phase-space point of view, based on Wigner function, is adopted, which provides a quasi-classical description of the quantum dynamics. After a theoretical introduction to the Wigner-Weyl theory, a two-band phase-space transport model is developed, as an example of application of the Wigner formalism to the k⋅p framework. The third part of the chapter is devoted to quantum-fluid models, which are formulated in terms of a finite number of macroscopic moments of the Wigner function. For mixed-states, the maximum-entropy closure of the moment equations is discussed in general terms. Then, details are given on the multi-band case, where “multi-band” is to be understood in the wider sense of “multi-component wave function”, including therefore the case of particles with spin or spin-like degrees of freedom. Three instances of such systems, namely the two-band k⋅p model, the Rashba spin-orbit system and the graphene sheet, are examined.

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Notes

  1. 1.

    Equation (1.36) has been rewritten here with a slightly different notation. Moreover the mass is set equal to 1

  2. 2.

    A rigorous proof of existence and uniqueness of the constrained minimization problem has been recently obtained by Méhats and Pinaud [78, 79] for the moments up to first degree (density and current).

  3. 3.

    Recalling definition (1.50), a comparison with (1.48) shows that the physical density and velocity are given by N 0 n and p 0 u, where N 0 = (p 0∕2π ℏ)2.

  4. 4.

    In dimensionless variables, all the identities involving Weyl quantization are obtained from the original ones by the formal substitution ℏ↦ε.

  5. 5.

    Of course, there are also examples of multi-band QFD equations with an arbitrary number of bands, see e.g. [96].

  6. 6.

    For notational convenience we adopt “+” and “−” to denote “up” and “down” instead of the more common “ ↑” and “ ↓”.

  7. 7.

    This shows clearly that, indeed, p is not a momentum but a “pseudomomentum” (or “crystal momentum”): the latter has rather different properties.

References

  1. E.N. Adams, Motion of an electron in a perturbed periodic potential. Phys. Rev. 85, 41–50 (1952)

    MATH  Google Scholar 

  2. G. Alì, G. Mascali, V. Romano, R.C. Torcasio, A hydrodynamic model for covalent semiconductors with applications to GaN and SiC. Acta Applicandae Mathematicae (2012)

    Google Scholar 

  3. G. Alì, D. Bini, S. Rionero, Global Existence and Relaxation Limit for Smooth Solutions to the Euler-Poisson Model for Semiconductors. SIAM J. Math. Anal. 32 3, 572–587 (2000)

    MATH  MathSciNet  Google Scholar 

  4. T. Ando, H. Akera, Connection of envelope functions at semiconductor heterointerfaces. II. Mixings of Γ and X valleys in GaAs/Al x Ga1−x As. Phys. Rev. B 40(17), 11619–11633 (1989)

    Google Scholar 

  5. L.C. Andreani, A. Pasquarello, F. Bassani, Hole subbands in strained GaAs-Ga1−x Al x -As quantum wells: Exact solution of the effective-mass equation, Phys. Rev. B 36(11), 5887–5894 (1987)

    Google Scholar 

  6. A.M. Anile, O. Muscato, Improved hydrodynamical model for carrier transport in semiconductors. Phys. Rev. B 51(23), 16728–16740 (1995)

    Google Scholar 

  7. A.M. Anile, V. Romano, Non parabolic band transport in semiconductors closure of the moment equations. Cont. Mech. Thermod. 11, 307–325 (1999)

    MATH  MathSciNet  Google Scholar 

  8. A. Arnold, C. Ringhofer An operator splitting method for the Wigner-Poisson problem. SIAM J. Numer. Anal. 33(4) 1622–1643 (1996)

    MathSciNet  Google Scholar 

  9. A. Arnold, Self-consistent relaxation-time models in quantum mechanics. Commun. Part. Diff. Eqs 21(3–4), 473–506 (1996)

    MATH  Google Scholar 

  10. C. Auer, F. Schürrer, Semicontinuous kinetic theory of the relaxation of electrons in GaAs. Transport theory and statistical physics 33, 429–447 (2004)

    Google Scholar 

  11. C. Auer, A. Majorana, F. Schürrer, Numerical schemes for solving the non-stationary Boltzmann-Poisson system for two-dimensional semiconductor devices. In: ESAIM Proceedings 15, 75–86 (2005)

    MATH  Google Scholar 

  12. C. Auer, F. Schürrer, C. Ertler, Hot phonon effects on the high-field transport in metallic carbon nanotubes. Phys. Rev. B 74(16), 165409 (2006)

    Google Scholar 

  13. T.B. Bahder, Eight-band k⋅ p model of strained zinc-blende crystals, Phys. Rev. B 41(17), 11992–12001 (1991)

    Google Scholar 

  14. L. Barletti, Wigner envelope functions for electron transport in semiconductor devices. Transp. Theory Stat. Phys. 32(3/4), 253–277 (2003)

    MATH  MathSciNet  Google Scholar 

  15. L. Barletti, N. Ben Abdallah, Quantum transport in crystals: effective-mass theorem and k⋅ p Hamiltonians. Commun. Math. Phys. 307, 567–607 (2011)

    MATH  MathSciNet  Google Scholar 

  16. L. Barletti, C. Cintolesi, Derivation of isothermal quantum fluid equations with Fermi-Dirac and Bose-Einstein statistics. J. Stat. Phys. 148 353–386 (2012)

    MATH  MathSciNet  Google Scholar 

  17. L. Barletti, G. Frosali, Diffusive limit of the two-band k⋅ p model for semiconductors. J. Stat. Phys. 139(2), 280–306 (2010)

    MATH  MathSciNet  Google Scholar 

  18. L. Barletti, G. Frosali, L. Demeio L.: multi-band quantum transport models for semiconductor devices. In: C. Cercignani, E. Gabetta (eds.), Transport Phenomena and Kinetic Theory, Modeling and Simulation in Science, Engineering and Technology, pp. 55–89. Birkhäuser, Boston (2007)

    Google Scholar 

  19. L. Barletti, F. Méhats, Quantum drift-diffusion modeling of spin transport in nanostructures. J. Math. Phys. 51(5), 053304 (2010)

    Google Scholar 

  20. G. Bastard, Wave mechanics applied to semiconductor heterostructures (Les Editions de Physique, Halsted Press, Les Ulis Cedex, 1988)

    Google Scholar 

  21. N. Ben Abdallah, On a multidimensional Schrödinger-Poisson scattering model for semiconductors. J. Math. Phys. 41(7), 4241–4261 (2000)

    MATH  MathSciNet  Google Scholar 

  22. I. Bialynicki-Birula, Hydrodynamic form of the Weyl equation. Acta Physica Polonica 26(7), 1201–1208 (1995)

    MATH  MathSciNet  Google Scholar 

  23. Y. Bychkov, E.I. Rashba, Properties of a 2D electron gas with lifted spectral degeneracy. JETP Lett. 39(2), 78–81 (1984)

    Google Scholar 

  24. P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94, 511–525 (1954)

    MATH  Google Scholar 

  25. D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden variables” I. Phys. Rev. 85, 166–179 (1952)

    MATH  MathSciNet  Google Scholar 

  26. D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden variables” II. Phys. Rev. 85, 180–193 (1952)

    MathSciNet  Google Scholar 

  27. L.L. Bonilla, L. Barletti, M. Alvaro, Nonlinear electron and spin transport in semiconductor superlattices, SIAM. J. Appl. Math. 69(2) 494–513 (2008)

    MATH  MathSciNet  Google Scholar 

  28. L.L. Bonilla, H.T. Grahn, Non-linear dynamics of semiconductor superlattices, Rep. Prog. Phys. 68 577–683 (2005)

    Google Scholar 

  29. G. Borgioli, G. Frosali, P. Zweifel, Wigner approach to the two-band Kane model for a tunneling diode. Transp. Theory Stat. Phys. 32(3/4), 347–366 (2003)

    MATH  MathSciNet  Google Scholar 

  30. M.G. Burt, The justification for applying the effective-mass approximation to microstructure. J. Phys: Condens. Matter, 4, 6651–6690 (1992)

    Google Scholar 

  31. C. Brouder, G. Panati, M. Calandra, C. Mourougane, N. Marzari, Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007)

    Google Scholar 

  32. F. Buot, K. Jensen, Lattice Weyl-Wigner formulation of exact many-body quantum-transport theory and applications to novel solid-state quantum-based devices. Phys. Rev. B 42, 9429–9457 (1990)

    Google Scholar 

  33. V.D. Camiola, G. Mascali, V. Romano, Numerical simulation of a double-gate MOSFET with a subband model for semiconductors based on the maximum entropy principle. Cont. Mech. Thermodyn. 70, 710 (2011)

    Google Scholar 

  34. P. Carruthers, F.C. Zachariasen, Quantum Collision Theory with Phase-Space Distributions. Rev. Mod. Phys. 55(1), 245–285 (1983)

    MathSciNet  Google Scholar 

  35. C.Y.-P. Chao, S.L. Chuang, Resonant tunneling of holes in the multi-band effective-mass approximation. Phys. Rev. B 43(9), 7027–7039 (1991)

    Google Scholar 

  36. S.L. Chuang, Efficient band-structure calculations of strained quantum wells. Phys. Rev. B 43(12), 9649–9661 (1991)

    MathSciNet  Google Scholar 

  37. J. des Cloizeaux, Energy bands and projection operators in a crystal: Analytic and asymptotic properties. Phys. Rev. 135(3A), A685–A697 (1964)

    Google Scholar 

  38. S. Datta, B. Das, Electronic analog of the electro-optic modulator, Appl. Phys. Lett. 56(7), 665–667 (1990)

    Google Scholar 

  39. Davidovich, M.A., Anda, E.V., Tejedor, C., Platero, G.: Interband resonant tunneling and transport in InAs/AlSb/GaSb heterostructures, Phys. Rev. B, 47(8), 4475–4484 (1993)

    Google Scholar 

  40. P. Degond, S. Gallego, F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation. Multiscale Model. Simul. 6(1), 246–272 (2007)

    MATH  MathSciNet  Google Scholar 

  41. P. Degond, C. Ringhofer, Quantum moment hydrodynamics and the entropy principle. J. Stat. Phys. 112(3–4), 587–628 (2003)

    MATH  MathSciNet  Google Scholar 

  42. P. Degond, F. Méhats, C. Ringhofer, Quantum energy-transport and drift-diffusion models. J. Stat. Phys. 118(3–4), 625–667 (2005)

    MATH  MathSciNet  Google Scholar 

  43. L. Demeio, L. Barletti, A. Bertoni, P. Bordone, C. Jacoboni, Wigner-function approach to multi-band transport in semiconductors. Physica B 314, 104–107 (2002)

    Google Scholar 

  44. L. Demeio, P. Bordone, C. Jacoboni, Numerical simulation of an intervalley transition by the Wigner-function approach. Semicond. Sci. Technol. 19, 1–3 (2004)

    Google Scholar 

  45. Di Stefano, V.: Modeling thermal effects in submicron semiconductor devices, Commun Appl. Industr. Math. 1(1), 110–117 (2010),

    Google Scholar 

  46. D. Dürr, S. Teufel, Bohmian mechanics. The physics and mathematics of quantum theory (Springer-Verlag, Berlin, 2009)

    Google Scholar 

  47. G.B. Folland, Harmonic Analysis in Phase Space (Princeton University Press, Princeton, 1989)

    MATH  Google Scholar 

  48. B.A. Foreman, Exact effective-mass theory for heterostructures. Phys. Rev. B 52, 12241–12259 (1995)

    Google Scholar 

  49. B.A. Foreman, Envelope-function formalism for electrons in abrupt heterostructures with material-dependent basis functions. Phys. Rev. B 54(3), 1909–1921 (1996)

    MathSciNet  Google Scholar 

  50. M. Freitag, Graphene: Nanoelectronics goes flat out. Nature Nanotechnology 3, 455–457 (2008)

    Google Scholar 

  51. W.R. Frensley, Boundary conditions for open quantum systems far from equilibrium. Rev. Mod. Phys. 62, 745–791 (1990)

    Google Scholar 

  52. G. Frosali, O. Morandi, A quantum kinetic approach for modeling a two-band resonant tunneling diode. Transp. Theory Stat. Phys. 36, 159–177 (2007)

    MATH  Google Scholar 

  53. S. Gallego, F. Méhats, Numerical approximation of a quantum drift-diffusion model. C.R. Math. Acad. Sci. Paris 339(7), 519–524 (2004)

    Google Scholar 

  54. S. Gallego, F. Méhats, Entropic discretization of a quantum drift-diffusion model. SIAM J. Numer. Anal. 43(5), 1828–1849 (2005)

    MATH  MathSciNet  Google Scholar 

  55. M. Galler, F. Schürrer, A deterministic solution method for the coupled system of transport equations for the electrons and phonons in polar semiconductors, J. Phys. A: Math. Gen. 37 1479–1497 (2004)

    MATH  Google Scholar 

  56. I. Gasser, P.A. Markowich, A. Unterreiter, Quantum hydrodynamics. In: P.A. Raviart (ed.) Modeling of Collisions, pp. 179–216. Gauthier-Villars, Paris (1997)

    Google Scholar 

  57. J. Genoe, K. Fobelets, C. van Hoof, G. Borghs, In-plane dispersion relations of InAs/AlSb/GaSb/AlSb/InAs interband resonant-tunneling diodes. Phys. Rev. B 52(19), 14025–14034 (1995)

    Google Scholar 

  58. H. Haug, S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (4th Edition, World Scientific, Singapore, 2004)

    Google Scholar 

  59. G.J. Iafrate, J.B. Krieger, Quantum transport for Bloch electrons in inhomogeneous electric fields. Phys. Rev. B 40(9), 6144–6148 (1989)

    Google Scholar 

  60. K.L. Jensen, F.A. Buot, Numerical simulation of intrinsic bistability and high-frequency current oscillations in resonant tunneling structures. Phys. Rev. Lett. 66, 1078–1081 (1991)

    Google Scholar 

  61. A. Jüngel, D. Matthes, A derivation of the isothermal quantum hydrodynamic equations using entropy minimization. Z. Angew. Math. Mech. 85(11), 806–814 (2005)

    MATH  MathSciNet  Google Scholar 

  62. M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Chiral tunnelling and the Klein paradox in graphene. Nature Physics 2(9), 620–625 (2006)

    Google Scholar 

  63. E.O. Kane, Energy band structure in p-type Germanium and Silicon. J. Phys. Chem. Solids 1, 82–89 (1956)

    Google Scholar 

  64. W. Kohn, Analytic Properties of Bloch Waves and Wannier Functions. Phys. Rev. 115, 809–821 (1959)

    MATH  MathSciNet  Google Scholar 

  65. D.Y.K. Ko, J.C. Inkson, Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems. Phys. Rev. B 38, 9945–9951 (1988)

    Google Scholar 

  66. H. Kosina, M. Nedjalkov, Review Chapter: Wigner Function Based Device Modeling. In: Handbook of Theoretical and Computational Nanotechnology (Los Angeles), 731–763 (2006)

    Google Scholar 

  67. H.-W. Lee, Theory and application of the quantum phase-space distribution functions, Phys. Rep. 259, 147–211 (1995)

    MathSciNet  Google Scholar 

  68. L. Leibler, Effective-mass theory for carriers in graded mixed semiconductors. Phys. Rev. B 12 4443–4451 (1975)

    Google Scholar 

  69. P. Lichtenberger, O. Morandi, F. Schürrer, High field transport and optical phonon scattering in graphene, Phys. Rev. B 84, 045406 (2011)

    Google Scholar 

  70. J.M. Luttinger, W. Kohn, Motion of electrons and holes in perturbed periodic fields. Phys. Rev. 97(4), 869–882 (1955)

    MATH  Google Scholar 

  71. E. Madelung, Quantentheorie in hydrodynamischer Form. Zeitschr. f. Phys. 40, 322–326 (1926)

    MATH  Google Scholar 

  72. R.K. Mains, G.I. Haddad, Wigner function modeling of resonant tunneling diodes with high peak-to-valley ratios. J. Appl. Phys. 64, 5041–5044 (1988)

    Google Scholar 

  73. C. Manzini, L. Barletti, An analysis of the Wigner-Poisson problem with inflow boundary conditions. Nonlinear Analysis, 60(1), 77–100 (2005)

    MATH  MathSciNet  Google Scholar 

  74. N. Marzari, D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847–12865 (1997)

    Google Scholar 

  75. G. Mascali, V. Romano, A non parabolic hydrodynamical subband model for semiconductors based on the maximum entropy principle. Math. Comput. Modeling 55(3–4), 1003–1020 (2012)

    MATH  MathSciNet  Google Scholar 

  76. F. Méhats, Adiabatic Approximation of the Schrödinger–Poisson System with a Partial Confinement. SIAM J. Math. Anal. 36(3), 986–1013 (2005)

    MATH  Google Scholar 

  77. F. Méhats, Analysis of a Quantum Subband Model for the Transport of Partially Confined Charged Particles. Monatsh. Math. 147, 43–73 (2006)

    MATH  MathSciNet  Google Scholar 

  78. F. Méhats, O. Pinaud, An inverse problem in quantum statistical physics. J. Stat. Phys. 140, 565–602 (2010)

    MATH  MathSciNet  Google Scholar 

  79. F. Méhats, O. Pinaud, A problem of moment realizability in quantum statistical physics. Kinet. Relat. Models 4(4), 1143–1158 (2011)

    MATH  MathSciNet  Google Scholar 

  80. A.T. Meney, B. Gonul, E.P. O’Reilly, Evaluation of various approximations used in the envelope-function method. Phys. Rev. B 50(15), 10893–10904 (1994)

    Google Scholar 

  81. O. Morandi, multi-band Wigner-function formalism applied to the Zener band transition in a semiconductor. Phys. Rev. B 80, 024301 (2009)

    Google Scholar 

  82. O. Morandi, Effective classical Liouville-like evolution equation for the quantum phase space dynamics. J. Phys. A: Math. Theor. 43 365302 (2010)

    MathSciNet  Google Scholar 

  83. O. Morandi, Quantum Phase-Space Transport and Applications to the Solid State Physics. In M.R. Pahlavani (ed.), Some Applications of Quantum Mechanics, 1–26. InTech, (2012)

    Google Scholar 

  84. O. Morandi, M. Modugno, A multi-band envelope function model for quantum transport in a tunneling diode. Phys. Rev. B 71 235331 (1–8) (2005)

    Google Scholar 

  85. R.A. Morrow, K.R. Brownstein, Model effective-mass Hamiltonians for abrupt heterojunctions and the associated wave-function-matching conditions. Phys. Rev. B 30, 678–680 (1984)

    Google Scholar 

  86. J.E. Moyal, Quantum mechanics as a statistical theory. Math. Proc. Cambr. Philos. Soc. 45, 99–124 (1949)

    MATH  MathSciNet  Google Scholar 

  87. O. Muscato, Validation of an Extended Hydrodynamical model for a submicron npn Bipolar Junction Transistor. Physica A 365(2), 409–428 (2006)

    Google Scholar 

  88. O. Muscato, Hot electron distribution function for the Boltzmann equation with analytic bands. J. Comput. Electron. 5, 377–380 (2006)

    MathSciNet  Google Scholar 

  89. O. Muscato, V. Di Stefano, Modeling heat generation in a sub-micrometric n+ - n - n+ silicon diode. J. Appl. Phys.104, 124501 (2008)

    Google Scholar 

  90. M. Nedjalkov, D. Querlioz, P. Dollfus, H. Kosina, Review Chapter: Wigner Function Approach. In: Nano-Electronic Devices: Semiclassical and Quantum Transport Modeling, pp. 1–76, Springer-Verlag, Berlin (2011)

    Google Scholar 

  91. G. Nenciu, Existence of the exponentially localised Wannier functions. Comm. Math. Phys. 91, 81–85 (1983)

    MATH  MathSciNet  Google Scholar 

  92. M. Ogawa, R. Tominaga, T. Miyoshi, Multi-Band Simulation of Interband Tunneling Devices Reflecting Realistic Band Structure. IEICE Trans. Electr. E83-C(8) 1235–1241 (2000)

    Google Scholar 

  93. D. Querlioz, P. Dollfus, The Wigner Monte Carlo Method for Nanoelectronic Devices – A particle description of quantum transport and decoherence. ISTE-Wiley, (2010)

    Google Scholar 

  94. G. Panati, A. Pisante, Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions. arXiv:1112.6197 [math-ph] (2011)

    Google Scholar 

  95. C. Reitshammer, F. Schürrer, A. Rossani, Atomic Beam Slowing and Cooling: Discrete Velocity Model. Phys. Rev. E. 58 3964–3970 (1998)

    Google Scholar 

  96. C. Ringhofer, Sub-band diffusion models for quantum transport in a strong force regime. SIAM J. Appl. Math. 71(6), 1871–1895 (2011)

    MATH  MathSciNet  Google Scholar 

  97. V. Romano, Non parabolic band transport in semiconductors: closure of the production terms in the moment equations. Cont. Mech. Thermodyn. 12, 31–51 (1999)

    MathSciNet  Google Scholar 

  98. V. Romano, Non-parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices. Math. Meth. Appl. Sci. 24, 439–471 (2001)

    MATH  MathSciNet  Google Scholar 

  99. L. Shifren, C. Ringhofer, D.K. Ferry, A Wigner Function-Based Quantum Ensemble Monte Carlo Study of a Resonant Tunneling Diode. IEEE Trans. Electron Devices 50(3), 769–773 (2003)

    Google Scholar 

  100. J.C. Slonczewski, P.R. Weiss, Band structure of graphite. Phys. Rev. 109(2), 272–279 (1958)

    Google Scholar 

  101. M. Sweeney J.M. Xu, Resonant interband tunnel diodes. Appl. Phys. Lett. 54(6), 546–548 (1989)

    Google Scholar 

  102. D. Taj, L. Genovese, F. Rossi, Quantum-transport simulations with the Wigner-function formalism: failure of conventional boundary-condition schemes. Europhys. Lett. 74(6), 1060–1066 (2006)

    Google Scholar 

  103. T. Takabayasi, The vector representation of spinning particle in the quantum theory. Prog. Theor. Phys. 14(4), 283–302 (1955)

    MATH  MathSciNet  Google Scholar 

  104. B. Thaller, The Dirac Equation (Springer-Verlag, Berlin, 1992)

    Google Scholar 

  105. S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics (Springer-Verlag, Berlin, 2003)

    MATH  Google Scholar 

  106. S. Teufel, G. Panati, Propagation of Wigner functions for the Schrödinger equation with a perturbed periodic potential. arXiv:math-ph/0403037 (2004)

    Google Scholar 

  107. M. Trovato, L. Reggiani, Quantum maximum entropy principle for a system of identical particles. Phys. Rev. E 81, 021119 (2010)

    Google Scholar 

  108. M. Trovato, L. Reggiani, L.: Quantum hydrodynamic models from a maximum entropy principle. J. Phys. A: Math. Theor. 43, 102001 (2010)

    Google Scholar 

  109. M. Trovato, L. Reggiani, Maximum entropy principle and hydrodynamic models in statistical mechanics. La Rivista del Nuovo Cimento 35, 99–266 (2012)

    Google Scholar 

  110. R. Tsu, L. Esaki, Tunneling in a finite superlattice. Appl. Phys. Lett. 22(11), 562–564 (1973)

    Google Scholar 

  111. M.B. Unlu, B. Rosen, H.-L. Cui, P. Zhao, Multi-band Wigner function formulation of quantum transport. Phys. Lett. A 327(2–3), 230–240 (2004)

    MATH  Google Scholar 

  112. C.M. van Vliet, A.H. Marshak, Wannier-Slater theorem for solid with nonuniform band structure. Phys. Rev. B 26(12), 6734–6738 (1982)

    Google Scholar 

  113. J. Von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955)

    MATH  Google Scholar 

  114. G.H. Wannier, Dynamics of band electrons in electric and magnetic fields. Rev. Mod. Phys. 34, 645–655 (1962)

    MathSciNet  Google Scholar 

  115. W.T. Wenckebach, Essential of Semiconductor Physics (John Wiley & Sons, Chichester, 1999)

    Google Scholar 

  116. G.Y. Wu, K.-P. Wu, Electron transport in a resonant-tunneling diode under the effect of a transverse magnetic field: a quantum theory in the Wigner formalism. J. Appl. Phys. 71(3), 1259–1264 (1992)

    Google Scholar 

  117. E. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)

    Google Scholar 

  118. R.Q. Yang, M. Sweeny, D. Day, J.M. Xu, Interband tunneling in heterostructure tunnel diodes. IEEE Trans. Electron Devices 38(3), 442–446 (1991)

    Google Scholar 

  119. N. Zamponi, Some fluid-dynamic models for quantum electron transport in graphene via entropy minimization. Kinet. Relat. Mod. 5(1), 203–221 (2012)

    MATH  MathSciNet  Google Scholar 

  120. N. Zamponi, L. Barletti, Quantum electronic transport in graphene: a kinetic and fluid-dynamical approach. Math. Meth. Appl. Sci. 34, 807–818 (2011)

    MATH  MathSciNet  Google Scholar 

  121. N. Zamponi, A. Jüngel, Two spinorial drift-diffusion models for quantum electron transport in graphene. Comm. Math. Sci. (in press)

    Google Scholar 

  122. I. Žutić, J. Fabian, S. Das Sarma, Spintronics: fundamentals and applications. Rev. Mod. Phys. 76(2), 323–410 (2002)

    Google Scholar 

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Acknowledgements

This work was partially supported by the Italian Ministry of University (MIUR National Project “Kinetic and hydrodynamic equations of complex collisional systems”, PRIN 2009, Prot. n. 2009NAPTJF_003).

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  1. Dipartimento di Matematica e Informatica “U.Dini”, Università di Firenze, Viale Morgagni 67/a, 50134, Firenze, Italy

    Luigi Barletti & Giovanni Frosali

  2. Institut de Physique et Chimie des Matériaux de Strasbourg, Université de Strasbourg, CNRS UMR 7504, 23 rue du Loess, BP 43, 67034, Strasbourg Cedex 2, France

    Omar Morandi

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Correspondence to Giovanni Frosali .

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  1. & Numerische Mathematik, Bergische Universität Wuppertal Lehrstuhl für Angewandte Mathematik, Wuppertal, Germany

    Matthias Ehrhardt

  2. Forschungsgruppe “Partielle Differentialgleichungen", Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany

    Thomas Koprucki

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Barletti, L., Frosali, G., Morandi, O. (2014). Kinetic and Hydrodynamic Models for Multi-Band Quantum Transport in Crystals. In: Ehrhardt, M., Koprucki, T. (eds) Multi-Band Effective Mass Approximations. Lecture Notes in Computational Science and Engineering, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-01427-2_1

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